Equations of some embeddings of a projective space into another one

In arXiv:math/0405373 , Eisenbud, Huneke and Ulrich conjectured a result on the Castelnuovo-Mumford regularity of the embedding of a projective space $\mathbb{P}^{n-1}\hookrightarrow \mathbb{P}^{r-1}$ determined by generators of a linearly presented $\mathfrak{m}$-primary ideal. This result implies in particular that the image is scheme defined by equations of degree at most $n$. In this text we prove that the ideal of maximal minors of the Jacobian dual matrix associated to the input ideal defines the image as a scheme; it is generated in degree $n$. Showing that this ideal has a linear resolution would imply that the conjecture in arXiv:math/0405373 holds. Furthermore, if this ideal of minors coincides with the one of the image in degree $n$ - what we hope to be true - the linearity of the resolution of this ideal of maximal minors is equivalent to the conjecture in arXiv:math/0405373.